Horizon-scale images of M87* and Sgr A* are produced by a plasma that is collisionless, relativistic and magnetized, in a regime where the radiating electrons are energized by kinetic processes operating several orders of magnitude below the scales we observe. This gap is currently bridged by tunable prescriptions for electron heating, and the resulting degeneracies limit what the images can tell us about accretion and jet launching.

The first part of the talk reviews how relativistic turbulence and magnetic reconnection energize particles near a black hole, and how fully kinetic simulations are now being used to constrain electron heating from first principles.

All of this, however, rests on a step that is usually taken for granted: the statistical tools used to characterize turbulence near the horizon are inherited from flat spacetime. In the second part of the talk I will present a formalism for second-order structure functions and power spectral densities that is built consistently on a curved background. Applied to GRMHD accretion flows onto a Kerr black hole, it recovers the inertial cascade but departs from the flat-spacetime estimate by 40–80% close to the horizon. I will then show what an inhomogeneous accretion flow leaves as a spectral signature, and close with an assessment of what is still missing.